In this section, we first prove Theorem REF  (ii) and (iii) (recall that (i) is already known, see [1]}) and then state and prove an analogous result for walks attracted to a stable law as alluded to in the introduction.
Recall the framework of Sec. REF : \(\xi \)  is a random variable supported by \(\mathbb {Z}_{\geqslant 0}\) , and, in order to avoid periodicity issues, we assume throughout that its support is not included in a sublattice of \(\mathbb {Z}\) . We consider a random walk, say \(S_{n} = \xi _{1} + \dots + \xi _{n}\)  for every \(n \geqslant 1\) , where \((\xi _{i})_{i\geqslant 1}\)  is a sequence of i.i.d. copies of \(\xi \) .
Recall that we denote by \(G(s) = \sum _{k \geqslant 0} s^{k} \mathbb {P}(\xi =k)\)  the generating function of \(\xi \) , and we let \(\rho \geqslant 1\)  denote its radius of convergence. For every \(b \in (0, \rho )\) , let \(\xi ^{(b)}\)  have the law with generating function \(G^{(b)}(s) = G(bs)/G(b)\) , so \(\mathbb {E}[\xi ^{(b)}] = b G^{\prime }(b)/G(b)\)  and let \(S^{(b)}\)  denote the associated random walk.
